# The hyperbolicity constant of infinite circulant graphs

Open Mathematics (2017)

• Volume: 15, Issue: 1, page 800-814
• ISSN: 2391-5455

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## Abstract

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If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

## How to cite

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José M. Rodríguez, and José M. Sigarreta. "The hyperbolicity constant of infinite circulant graphs." Open Mathematics 15.1 (2017): 800-814. <http://eudml.org/doc/288465>.

@article{JoséM2017,
abstract = {If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = \{x1, x2, x3\} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.},
author = {José M. Rodríguez, José M. Sigarreta},
journal = {Open Mathematics},
keywords = {Circulant graph; Gromov hyperbolicity; Geodesics; Infinite graphs},
language = {eng},
number = {1},
pages = {800-814},
title = {The hyperbolicity constant of infinite circulant graphs},
url = {http://eudml.org/doc/288465},
volume = {15},
year = {2017},
}

TY - JOUR
AU - José M. Rodríguez
AU - José M. Sigarreta
TI - The hyperbolicity constant of infinite circulant graphs
JO - Open Mathematics
PY - 2017
VL - 15
IS - 1
SP - 800
EP - 814
AB - If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.
LA - eng
KW - Circulant graph; Gromov hyperbolicity; Geodesics; Infinite graphs
UR - http://eudml.org/doc/288465
ER -

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